If the veneer is open grain, clean it with the thinner used in open grain finishes such as Watco Danish Oil. I believe that's mineral spirits. Check to be sure.
Dampen a clean cloth with the thinner, and wipe the surface of the veneer. It should clean things up. If the veneer really has an open grain finish, the thinner will wet and temporarily darken the veneer. It may even make the grain show up better. As the thinner dries, it should return to it's paler original appearance. If you see that, then you know you have an open grain finish. If you like, after cleaning and a very light sanding with fine sandpaper or 0000 steel wool, apply a coat of clear Watco Danish Oil. It will penetrate the wood grain and dry, leaving a coat of linseed oil, but maintaining the open grain.
If the veneer is sealed, such as with a polyurethane varnish, the thinner should have little effect on the wood grain color. You can clean it mild soap and water. After that, you may want to go with light sanding to remove scratches, followed by another coat of poly. You'll have to decide if you want glossy, satin, or matte poly.